The Hawks post-season berth comes after a 25-11 win-loss ratio under interim coach Nate McMillan.
Russell Westbrook had 34 points, 15 assists but only five rebounds for Washington, two nights after breaking Oscar Roberson’s record with his 182nd career triple-double.
DeMar DeRozan had 21 points for the Spurs, who missed a chance to secure a spot in the play-in tournament.
Luka Doncic scored 33 points to lead the Dallas Mavericks to a 125-107 win over the visiting New Orleans Pelicans.
Kristaps Porzingis scored 19 on his return from injury in a win which means Dallas can clinch at least the sixth seed in the Western Conference play-offs by winning one of their two remaining fixtures.
The undermanned Pelicans, restricted by injuries to only four regular starters, were led by Eric Bledsoe and Jaxson Hayes with 15 points each as they were eliminated from post-season contention.
Elsewhere, the Portland Trail Blazers upset the Utah Jazz 105-98 in Salt Lake City, the Cleveland Cavaliers beat the visiting Boston Celtics 102-94 and the Los Angeles Lakers edged past the visiting Houston Rockets 124-122.